A successful private Blockchain experiment in Burma might change the way microfinance works in developing nations.
Developing countries present a huge opportunity which is hidden under veil of a problem in the form of the underbanked. Burma is one such emerging economy. In such countries providing banking services to those who have been the most ignored by mainstream economy can bring about significant benefits.
In neighbouring Bangladesh, microfinance has been hailed as a great success and nearly 700 microfinance institutions are operational which have been disbursing Bangladesh Taka 647 billion to nearly 34 million active borrowers according to the think tank, Copenhagen Consensus.
According to the Consultative Group to Assist the Poor (CGAP), the demand for microfinance in Burma is four times the current supply and is one of the reasons why the country is finding it hard to develop as it becomes more open. It is, therefore, of vital importance that microfinance institutions in Burma receive support in the form of technological assistance and otherwise.
Recently Infoteria Corporation and the Tech Bureau Corporation of Japan have been working with BC Finance Holdings Limited on experiments which are based on Tech Bureau's Mijin Platform. Mijin is a private Blockchain placed on Microsoft Azure using ASTERIA WARP and Mijin adapters.
Using this technology Infoteria and Techbureau have been successful in transferring loan and deposit account data in the main system of Burma's BC finance. This demonstration was carried out between May 28 and June 5, 2016.
The experiment was successfully conducted as Infoteria and Techbureau could record all the transaction history of one of the 19 branches of BC Finance in the private Blockchain Mijin.
We asked Jeremy Kloiser-Jones, Founder and CEO of BC Finance Holdings Limited, about the successful test of Blockchain technology in a microfinance environment.
At the moment even in Burma the fintech landscape is rather limited. However, there are possibilities that are emerging due to the country embracing openness.
Mobile penetration has increased from 10 per cent four years ago, to 50 per cent at the moment. Mobile penetration can act as an important delivery channel for microfinance companies.
The Burmese economy is now growing at the rate of 8.5 per cent p.a. And, thus, there is a need for more of everything according to Jeremy Kloiser-Jones of BC Finance.
Takao Asayama, CEO of Tech Bureau Corp, whose platform is being used by BC Finance has his own take on developing countries and the potential Fintech applications. He thinks that most developing countries, including Burma, have been using legacy technologies to run their operations. He cites the example of BC Finance who uses Excel spreadsheets to manage loans and then pushes those to a database daily. He thinks to upgrade that to an enterprise financial system could cost millions of dollars and may need reliable system integrators from overseas.
Using Blockchain things can be relatively simpler.
While developed countries may present an easier landscape to innovate, existing regulations make it difficult to get things off the ground in some cases. Developing countries in this way are more exciting for Fintech companies.
That Blockchain technology has started to make a real impact on the lives of people in developing countries is a positive development. We asked both Jeremy Kloiser-Jones and Takao Asayama about the future possibilities of using their technology in developing nations.
Kloiser-Jones is of the view that there is a big future for this technology and he expects it to contribute greatly to BC Finance's ability in providing flexible services to a broad audience of the underbanked. He says that they have started in Myanmar, but have a number of other countries in their sights for expansion.
Asayama plans to introduce 'solid' smart contracts concentrating on simple asset exchange instead of complex program language, which reduces the probability of errors and failures.
The Japanese are making forays into global fintech and reaching out to other countries and this is quite evident from the case of Mijin.
Will the Japanese Ninja be able to take the world by surprise? Only time will tell.


The floating sacred shrine of Haji Ali Dargah holds great esteem among its followers. To know more on the history, importance, and how to reach Haji Ali Mosque, read on.
Haji Ali is a revered Dargah (tomb) of the Muslims. Located on the brink of the shoreline of Worli Bay, this sacred shrine stands tall amidst sparkling blue sea as a mark of remembrance and reverence to Pir Haji Ali Shah Bukhari (R.A.). The mosque, which is apparently one of the most popular landmarks in Mumbai, was built in the 19th century. Not just the Mumbaikars but people from different parts of the globe come here to pay their homage to Haji Ali. The tomb is a classic example of Indian Islamic architecture. The sepulcher of Haji Ali is enshrouded with a brocaded red and green sheet, supported by a delicate silver frame. The 500-yard-causeway that extends to the Arabian Sea is the major highlight of this striking shrine. The Dargah is a renowned pilgrimage site among the Muslims and receives thousands of devotees everyday who visit this place to offer prayers and get their wishes fulfilled. The marble pillars inside the main hall are decorated with multi-hued mirrors and engraved names of Allah, which adds on to the beauty and sanctity of the place.
Haji Ali Dargah was built in the year 1413. The mosque is named after a wealthy Muslim merchant Pir Haji Ali Shah, who renounced all his worldly belongings before embarking on a holy pilgrimage to Mecca. As per the legend, the Saint once bumped on a poor woman who was wailing on the road. He walked up to her and enquired the cause of her grief. The poor women told him that she had stumbled and fallen, apparently spilling the oil that she was carrying, and now she was scared of being brutally beaten by her husband. On hearing this, he requested the woman to take him to the spot where she lost the oil. On reaching the place, he bent and poked a finger into the soil and much to the woman's amazement, a stream of oil gushed out. The ecstatic woman filled her vessel with oil and went home. Later, Pir Haji Ali Shah Bukhari had a disturbing dream that he had injured Mother Earth by his act. Full of remorse, he soon fell ill and requested his followers to cast the coffin carrying his body into the Arabian Sea, once he died. Haji Ali left this world during his journey to Mecca and miraculously the casket carrying his body floated back to these shores, being stuck on the string of rocky islets just off the Worli shore. The construction of the mosque took place after this incident.
One of the major attractions of this place is the Worli Bay. The seashore absolutely lends a scenic look to this age-old shrine. Trailing down the long walkway that cuts through the sea and leads you to the Dargah, is a one-of-its-kind of an experience. There is a mosque attached to the mausoleum of Haji Ali. This monument has always been the ultimate attraction of Mumbai. The best time to visit the Dargah is during Eid-E-Milad-Un-Nabi festival and the Urs (death anniversary) of Pir Haji Ali Shah Bhukari, which is celebrated on the 16th of every Islamic month (17thshab). The trust performs the rituals as per Islamic traditions and distributes sweets and food to the devotees and the visitors. A special program of Milad and prayers is held in remembrance of the Urs of Pir Haji Ali Shah Bukhari. During Eid-E-Milad-un-Nabi, the birth anniversary of Prophet Mohamed, the trust organizes a function and puts in display the sacred relics of Prophet Mohamed after Namaaz-e-Zuhr.
One needs to take a flight to Chhatrapati Shivaji International Airport, Mumbai to reach this shrine. From the airport, you can avail the public transports like bus or rickshaw to reach the shrine.
There are local trains available to Mumbai. Haji Ali is well connected by local trains. On the Western line, you need to get off at Mahalakshmi Station or Mumbai Central Station, while on the Central (Main) line you need to get off at Byculla Station and take a bus or taxi to the Dargah.
B.E.S.T city buses are available to this mosque. One can easily take a bus to reach the place. If you find difficulty in making the rickshaws agree to take, bus is the next best option to you.
Women are advised to cover their head with a dupatta and maintain the code of dressing when visiting this place. Since the mosque is located on the seashore, during the monsoon one gets a feeling that the mosque is floating in the sea. Hence, it is also called as the 'floating mosque of Mumbai'.


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Every single one of the UK's most successful and profitable businesses holds regular board meetings to discuss strategy. So, we'd like to invite you to benefit from our exciting BoardView service.
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Smile and Wave: Minibook for Katy.
KT, you might read this before you get your present so be warned that if you keep scrolling down, you'll ruin your surprise. But, if you can't wait, you can peek.
I'm the worst at forgetting birthdays, as documented here. So, I decided to at least make it a point to try to get something in the mail on time.
Didn't happen! But, it did get started. And it'll be there very soon after your birthday. I'm just proud that I was able to finish it before the 12th.
So, Katy, I hope you feel celebrated on Sunday and have sweet moments with your bearded hubby, and that your doggies are extra nice. Happy 27th! Here's your present!
this is so amazing! great job!!
I love the book!! The colors are so bright and eye-opening. Plus, you just have a lot of talent and energy radiating from the pages.
I have a GIVEAWAY on my site if you want to take a peek.
Beautiful! She's a very lucky girl!
Oh, I just love this book! I have a lot of love, elsie stuff that I just don't want to use up, but I think that I would be okay to do something like this and keep in all together in a book! Thanks for the inspiration and what a lucky b'day girl your friend is!
This book looks awsome. I just got the new zutter bind it all machine and am so looking forward to creating books. The love elsie paper looks awsome in it.
What did you use to tie the book together?
that is soooo cool! awesome!


text:
\section{Introduction}

In a fixed projective space $\mathbb P^n$, let $Z$ be a zero-dimensional scheme.
We denote by $h_Z$ the Hilbert function of $Z$:
\[
h_Z(t) = \dim (R/I_Z)_t
\]
for all $t$.  We also use the analogous notation $h_{R/I_Z}(t)$ or, if $A$ is a
graded ring (e.g.\ Artinian), $h_A(t)$.

Consider the possible  zero-dimensional schemes in $\mathbb P^n$ having a fixed
Hilbert function, $H$.  Recall \cite{GMR} that the necessary and sufficient
condition for the existence of $H$ is that $H$ be a differentiable O-sequence
that stabilizes at some value $d$; i.e.\ not only $H$ but also its first
difference $\Delta H$ has to satisfy Macaulay's growth condition \cite{fsmacaulay}
(see the next section for a review), and $\Delta H (t) = 0$ for all sufficiently
large $t$.  Conversely, for any such function there always exists a reduced
zero-dimensional scheme with Hilbert function $H$.

We first review some  notation (mostly from \cite{RZ}).  We denote by
$\hhn$ the {\em postulation Hilbert scheme associated to $H$}, i.e.\ the Hilbert
scheme parameterizing zero-dimensional schemes with Hilbert function $H$. We
define $\hredn$ to be the open subscheme of $\hhn$ parameterizing the {\em
reduced} zero-dimensional schemes with Hilbert function $H$. If
$I$ is the ideal of a zero-dimensional scheme  such that
$R/I$ has Hilbert function $H$, suppose that $R/I$ has minimal free resolution
\begin{equation} \label{resol}
0 \rightarrow \bigoplus_j R^{\beta_{n,j}}(-j) \rightarrow \dots \rightarrow
\bigoplus_j R^{\beta_{1,j}}(-j) \rightarrow R \rightarrow R/I \rightarrow 0,
\end{equation}
with $\beta_{i,j} \in \mathbb N$.  We denote by $\beta^I$ the set of graded Betti
numbers of $R/I$ (or equivalently, the set of graded Betti
numbers of $I$).  We then write
\[
\mathbb B_H := \{ \beta^I : I \subset R \hbox{ and } h_{R/I} = H \},
\]
and
\[
\mathbb B'_H := \{ \beta^I : I \subset R , h_{R/I} = H \hbox{ and $I$ is
reduced} \}.
\]
$\mathbb B_H$ (resp.\ $\mathbb B'_H$) is a partially ordered set under the
inequality $\leq$, where we write $\beta^{I} \leq \beta^{I'}$ if $\beta_{i,j} \leq
\beta'_{i,j}$ (using the usual inequality for integers) for all $i,j$.

Two (related) questions may be asked, for a given Hilbert function $H$.  First, is
$\hredn$  reducible?   Second, what can we say about $\mathbb B_H$ and $\mathbb
B'_H$; and in particular, is there a unique minimum $\beta^I \in \mathbb B_H$
(resp.\ $\beta^I
\in \mathbb B'_H$) under the  partial ordering $\leq$? One can also ask similar
questions for Hilbert functions and graded Betti numbers of graded Artinian
algebras.

The two questions are related by the following theorem of Ragusa and Zappal\`a:

\begin{theorem}[\cite{RZ}] \label{RZ theorem}
If $\mathbb B_H$ (resp.\ $\mathbb B'_H$) has no unique minimum element then $\hhn$
(resp.\  $\hredn$) is reducible.
\end{theorem}

The point of this theorem is that the graded Betti numbers obey semicontinuity
for ideals in an irreducible family (cf.\ \cite{RZ}, Lemma 1.2), and within such
a family cancelation of Betti numbers can only come in consecutive terms in the
resolution (cf.\ \cite{peeva}).  So we can make the following rephrasing of
Theorem \ref{RZ theorem}: Let
$I$ and $I'$ be ideals of  zero-dimensional schemes with the same Hilbert
function,
$H$, and assume that
$\beta^I$ and $\beta^{I'}$ are incomparable under the partial ordering.  If no
ideal $J$ exists with $\beta^J$ obtainable both from $\beta^I$ and $\beta^{I'}$ by
consecutive cancelation, then $I$ and $I'$ correspond to different components
of  $\hhn$, which is thus reducible.  If $I$ and $I'$ are both reduced then
$\hredn$ is reducible.  We will use this idea without comment in the following
sections.

\begin{definition} \label{strongly incomparable def}
We say that the Betti diagrams of $I$ and $I'$ are {\em strongly incomparable} if
the conditions of the previous paragraph are satisfied.  \qed
\end{definition}

It is known
that $\hhtwo$ is smooth and irreducible \cite{gotzmann} and that
sometimes even $\hhn$  is irreducible \cite{gotzmann2}.  On the other hand,
recently there has been a great deal of activity in this area, giving examples of
many different kinds  to show that sometimes
$\hhn$ and $\hredn$ (or related objects in the Artinian situation) are reducible,
or that $\mathbb B_H$ has no minimum element (cf.\ for instance \cite{BV},
\cite{CE},
\cite{ER}, \cite{FR}, \cite{kleppe}, \cite{RZ}, \cite{RZ2}, \cite{richert},
\cite{rodriguez}).  Some of these examples are isolated, and some (e.g.\
\cite{BV},
\cite{RZ2}, \cite{richert}) are infinite families.  While we present a new large
class of Hilbert functions for which $\hredthree$ is reducible, the overlap with
any of these previous results is minimal.

To describe the results in this paper, we first recall some notions.

\begin{definition} An Artinian graded algebra, $A$, has the {\em Weak Lefschetz
Property (WLP)} if, for a general linear form $L$, the induced multiplication
$\times L : A_i \rightarrow A_{i+1}$ has maximal rank, for all $i$.
 A  zero-dimensional scheme is said to have WLP if its general Artinian
reduction does.  A reduced zero-dimensional scheme $Z$ has the {\em Uniform
Position Property (UPP)} if all subsets of the same cardinality have the same
Hilbert function.
\end{definition}

In the expository paper \cite{geomhilb}, the author discussed the Weak Lefschetz
Property (WLP) and the Uniform Position Property (UPP), describing both as open
conditions.  He showed in Example 3.4 that if $H$ is the Hilbert function with
first difference $(1,3,6,9,11,11,11)$ (corresponding to a set of 52 points in
$\mathbb P^3$), then $\hredthree$ is reducible.  Furthermore, in one component
no element has WLP, while in another component the general element has  WLP
(and in fact also UPP).

The purpose of this paper is to put this example into a  much more general
framework, in the process giving a much larger class of Hilbert functions for
which $\hredthree$ is reducible.  In addition, one component has general element
with WLP, and another has no element with WLP.  We also recall (see above) that
the Hilbert function, $H$, of a zero-dimensional scheme  is a differentiable
O-sequence.  To specify $H$, it is equivalent to specify its first difference,
$\Delta H$.  This is also known as the associated  {\em $h$-vector}.

Our main result is the following (cf.\ Corollary \ref{main corollary}):

\medskip

\noindent {\bf Theorem} {\em
Let $H$ be a Hilbert function in four variables, for which the first difference
is of the form
\[
\Delta H = \{ 1, 3, b_2, b_3, \dots, b_{t-1}, d, \dots, d, b_{s+2}, b_{s+3},
\dots , b_r, 0 \}.
\]
Here $t$ is simply the first degree for which the value is $=d$, and $s:= \lfloor
\frac{d-2}{2} \rfloor$.   We assume that
$(1,3,b_2,\dots,b_{t-1},d,\dots,d)$ is again a differentiable O-sequence, and
that
\[
\left \{
\begin{array}{ll}
b_{s+2} \leq d-2 & \hbox{if $d$ is even;} \\
b_{s+2} \leq d-1 & \hbox{if $d$ is odd}
\end{array}
\right.
\]
 and $b_i \geq b_{i+1}$ for all $i \geq
s+2$.  Assume furthermore that there are ``the right number" of $d$'s in the
middle. More precisely, we require that
\[
t \leq s-1 = \left \lfloor \frac{d-3}{2} \right \rfloor.
\]
 Then $\hredthree$ is reducible.  Furthermore, on one component no element has
WLP, while on another component the general element has WLP. }

\medskip

To put this in a different context, recall the following result:

\begin{theorem}[\cite{HMNW}, Proposition 3.5]
Let $\underline{h} = (1,b_1,b_2,\dots,b_r)$ be a finite sequence of positive
integers.  Then $\underline{h}$ is the Hilbert function of a graded Artinian
$k$-algebra $R/J$ having WLP if and only if $\underline{h}$ is a unimodal
O-sequence such that the positive part of the first difference is also an
O-sequence.
\end{theorem}

Notice that the conditions on $\Delta H$ in our result above are
very similar to this necessary and sufficient condition for WLP.  The
only differences are the requirement that we have ``the right number" of $d$'s,
and that
$b_{s+2} \leq d-2$ (instead of $d-1$) when $d$ is even.   However, the condition
on the right number of $d$'s can be restrictive -- see Example \ref{ci ex}, where
we also discuss how one might extend this result.  Nevertheless, we have the nice
fact that the first difference of the Hilbert function can begin with  any
differentiable O-sequence, and  end with any non-increasing sequence that goes to
zero.  This degree of freedom is surprising: it is not difficult to construct
different-looking sets of points with the same Hilbert function, but to show that
they have strongly incomparable Betti diagrams is difficult in general.

\section{Preliminaries}

Let $R = k[x_0,\dots,x_n]$, where $k$ is an algebraically closed field.
We first recall some basic facts about Hilbert functions, especially
those of arithmetically Cohen-Macaulay (ACM) curves in $\mathbb P^3$.

\begin{definition}
Let $Z \subset \mathbb P^{n-1}$ be any closed subscheme with defining (saturated)
ideal
$I = I_Z$.  We say that $Z$ is {\em arithmetically Cohen-Macaulay} (ACM) if the
coordinate ring $R/I_Z$ is a Cohen-Macaulay ring.  Note that if $Z$ is a
zero-dimensional scheme then it is automatically ACM. \qed
\end{definition}

\begin{definition} \label{ibinomexp}
The {\em $i$-binomial expansion} of the integer $a$ ($i, a >0$) is the unique
expression
\[
a = \binom{m_i}{i} + \binom{m_{i-1}}{i-1} + \dots + \binom{m_j}{j},
\]
where $m_i > m_{i-1} > \dots > m_j \geq j \geq 1$.  \qed
\end{definition}

\begin{definition}
If $a \in {\mathbb Z}$ ($a>0$) has $i$-binomial expansion as in Definition
\ref{ibinomexp}, then we set
\[
a^{\langle i \rangle} = \binom{m_i+1}{i+1} + \binom{m_{i-1}+1}{i} + \dots +
\binom{m_j+1}{j+1}.
\]
Note that this defines a collection of functions $^{\langle i \rangle}: \mathbb Z
\rightarrow \mathbb Z$.  \qed
\end{definition}

For example, the 5-binomial expansion of 76 is
\[
76 = \binom{8}{5}  + \binom{6}{4} + \binom{4}{3} + \binom{2}{2},
\]
so
\[
76^{\langle 5 \rangle} = \binom{9}{6} + \binom{7}{5} + \binom{5}{4} +
\binom{3}{3} = 111.
\]

\begin{definition} \label{o-seq}
A sequence of non-negative integers $\{ a_i : i \geq 0 \}$ is called an {\em
O-sequence} if
\[
a_0 = 1 \ \hbox{ and } \ a_{i+1} \leq a_i^{\langle i \rangle},
\]
for all $i$.  An O-sequence is said to have {\em maximal growth from degree $i$
to degree $i+1$} if $a_{i+1} = a_i^{\langle i \rangle}$.  \qed
\end{definition}

\begin{theorem}[\cite{fsmacaulay}] \label{macaulay thm}

The following are equivalent:

\begin{itemize}
\item[(i)] $\{ a_i : i \geq 0 \}$ is an O-sequence;

\item[(ii)] $\{ a_i : i \geq 0 \}$ is the Hilbert function of a standard graded
$k$-algebra.
\end{itemize}
\end{theorem}

\begin{definition}
Given a sequence of non-negative integers $ \underline{a} = $  $\{ a_i :
i \geq 0 \}$, the {\em first difference} of this sequence is the sequence $\Delta
\underline{a} := \{ b_i \}$ defined by $b_i = a_i - a_{i-1}$ for all $i$.  (We
make the convention that $a_{-1} = 0$, so $b_0 = a_0 = 1$.)  We say that
$\underline{a}$ is a {\em differentiable O-sequence} if $\Delta \underline{a}$ is
again an O-sequence.  By taking successive first differences, we inductively
define the {\em $k$-th difference}
$\Delta^k \underline{a}$.
\qed
\end{definition}

Recall that if $H = \{ h_i : i \geq 0 \}$ is the Hilbert function of an
arithmetically Cohen-Macaulay subscheme of $\mathbb P^n$ of dimension $r$ (so the
coordinate ring $R/I$ has Krull dimension $r+1$) then the sequences $\Delta^{i}
H$, $1 \leq i \leq r+1$, are all O-sequences.  We say that $H$ is $(r+1)$-times
differentiable.

In particular, let $H$ be the Hilbert function of a non-degenerate ACM curve,
$C$, of degree $d$ in $\mathbb P^3$.    Then $\Delta H$ is the Hilbert function
of a proper hyperplane section of $C$, which is a zero-dimensional scheme of
degree $d$ in the hyperplane $\mathbb P^2$.

Another result that we will need is the following.  Let $V$ be any reduced
subscheme of $\mathbb P^n$ (not necessarily ACM) with Hilbert function $H =
(1,n+1, h_2, h_3, \dots)$.  Let $e$ be any positive integer, and define a new
sequence
$\{ e_i : i \geq 0 \}$ by $e_i = \min \{ h_i, e \}$.  It is shown in \cite{GMR}
that
$\{ e_i \}$ is the Hilbert function of some reduced set of points on $V$.  The
sequence $\{ e_i \}$ is called {\em the truncation of $H$ at $e$}.

A useful way of presenting the graded Betti numbers occurring in the minimal free
resolution of a graded module is by the so-called {\em Betti diagram}, introduced
in the computer algebra program {\tt macaulay} \cite{macaulay}, a program which
was  used to generate many examples that contributed to the theorems in this
paper.  Specifically, for a standard graded algebra $R/I$ we have the diagram
\[
\begin{array}{cccccccccccccccccccccc}
0 & : & 1 & \beta_{1,1} & \beta_{2,2} & \beta_{3,3} \\
1 & : & - & \beta_{1,2} & \beta_{2,3} & \beta_{3,4} \\
2 & : & - & \beta_{1,3} & \beta_{2,4} & \beta_{3,5} \\
3 & : & - & \beta_{1,4} & \beta_{2,5} & \beta_{3,6} \\
\vdots && \vdots & \vdots & \vdots & \vdots
\end{array}
\]
Note that the minimal free resolution has redundant, or ``ghost'' terms if and
only if two non-zero entries are adjacent diagonal (with slope 1); for instance,
if $\beta_{2,5}$ and $\beta_{3,5}$ are both non-zero, then there is a copy of
$R(-5)$ in both the second and third free modules in the resolution.

\section{Basic Construction}

For most of the remainder of this paper we will work in the ring $R =
k[x_0,x_1,x_2,x_3]$ but we will also need the ring $S = k[x_1,x_2,x_3]$. The main
results of this paper are all built on the construction that we will describe in
this section.  There are certain invariants that we will need.

\begin{notation} \label{def of t}
Consider a twice differentiable O-sequence, $H = \{ h_i : i \geq 0 \}$ and assume
that  $h_1 = 4$ and that  $\Delta H$ levels off at the value $d$.  So $ H$
is the Hilbert function of an ACM curve, $C$,  in $\mathbb P^3$ and $\Delta H$ is
the Hilbert function of its general hyperplane section.

Let $\Delta H = \{ b_i : i \geq 0 \}$ and let $\Delta^2 H = \{ c_i : i \geq 0
\}$.  Then we define the integer $t$ by
\[
\Delta H = (1, 3, b_2, \dots, b_{t-1}, b_t = d, d, \dots)
\]
where $b_2 < b_3 < \dots < b_{t-1} < b_t = d$,
and we note that
\[
\Delta^2 H = (1,2, c_2,\dots,c_{t-1}, c_t, 0)
\]
with $c_t > 0$ and $\sum_{i=0}^t c_i = d$.
The integer
$t$ has been given many different names in the literature, and we have $\reg(I_C)
= t+1$.  The finite sequence $\Delta^2 H$ is also known as the {\em $h$-vector} of
$C$.

Now we set
\[
s :=  \left \lfloor \frac{d-1}{2} \right \rfloor
\]
and we let $CI$ be the Hilbert function of a complete intersection of type
$(2,s)$ in four variables.  That is,
\[
\Delta CI = (1,3,5, \dots, 2s-3, 2s-1, 2s) \ \hbox{ and } \ \Delta^2 CI =
(1,2,2,\dots,2,2,1)
\]
and note that $\Delta^2 CI$ ends in degree $s$.

We make the following key assumptions:
\begin{equation} \label{key assn}
d > 3 \ \hbox{ and } \ t \leq s-1 = \left  \lfloor \frac{d-3}{2} \right \rfloor.
\end{equation}
 \qed
\end{notation}

We will assume that $d$ is even.  The case $d$ odd is entirely analogous and
will be left to the reader (but we make comments from time to time about this
case).  Note that if $d$ is even then we have  $2s = d-2$.

\medskip

\noindent {\bf Claim 1:} $b_2 = 6$.

Suppose otherwise.  Recall that $H$ is twice differentiable, i.e.\ $\Delta^2 H$
is an O-sequence.  We have:
\begin{itemize}
\item $b_2 \geq 3$ since $\Delta H$ is  differentiable;

\item if $b_2 = 3$ then $d = 3$ (again using differentiablility), contradicting
our assumption;

\item if $b_2 = 4$ then $\Delta^2 H$ is the sequence
\[
1, 2, 1, \dots
\]
and by Macaulay's growth condition each subsequent entry must be 1 until degree
$t$, so
$d = t+2$, which clearly violates (\ref{key assn});

\item if $b_2 = 5$ then $\Delta^2 H$ is the sequence
\[
1,2,2,c_3,\dots,
\]
where each $c_i$ is $\leq 2$ and $c_{t+1} = 0$.  Hence $d \leq 2t+1$, so
combining with (\ref{key assn}), we get
\[
\frac{d-1}{2} \leq t \leq \left \lfloor \frac{d-3}{2} \right \rfloor,
\]
a contradiction.
\end{itemize}

\noindent Therefore we have finished Claim 1.

As noted, $CI$ is the Hilbert function of a complete intersection in $R$ of type
$(2, \frac{d-2}{2})$, $\Delta^2 CI$ ends in degree $\frac{d-2}{2} =
s$, and $2s = d-2$.  Consider the following table, where the last line is obtained
by taking the difference $\Delta H - \Delta CI$.

 {\scriptsize
\begin{equation} \label{table of calculations}
\begin{array}{l|ccccccccccccccccccccccccccccccccccc}
\deg & 0 & 1 & 2 & 3 & 4 & \dots & t-1 & t & t+1 & \dots  & s-2 & s-1 & s &
s+1 & \dots \\
\hline
\Delta H & 1 & 3 & 6 & b_3 & b_4 & \dots & b_{t-1} & d & d & \dots & d & d & d
& d & \dots \\
\Delta CI & 1 & 3 & 5 & 7 & 9 & \dots & 2t-1 & 2t+1 & 2t+3 & \dots  & 2s-3 &
2s-1 & 2s &  2s & \dots \\ \cline{2-16}
&&& 1 & e_1 & e_2 & \dots & e_{t-3} & e_{t-2} & e_{t-1} & \dots &  5 & 3 & 2 & 2 &
\dots
\end{array}
\end{equation}
}

\noindent Since $t \leq s-1$, we verify that the sequence $\{ e_i : i \geq 0 \}$
 ends with the subsequence \linebreak $(\dots, 3,2,2,2,\dots)$, although the
columns between $t$ and $s-1$ could be redundant.  (When $d$ is odd, the sequence
$\{ e_i : i \geq 0 \}$ ends with $(\dots, 2,1,1,1,\dots)$.)

\medskip

\noindent {\bf Claim 2:} {\em $\{ e_i : i \geq 0 \}$ is an O-sequence.  }

Note that
$e_1
\leq 3$.   We want to show that for any $i$, the growth from $e_i$ to $e_{i+1}$
does not violate Macaulay's theorem.

\medskip

\noindent {\underline{Case 1}}: $i \geq t-2$.

Then $i+2 \geq t$ so $b_{i+2} = d$ and it follows that $e_i \geq e_{i+1}$ (and in
fact $e_i > e_{i+1}$ for $t-2 \leq i \leq s-3$).  Hence $e_i$ is an O-sequence in
this range.

\medskip

\noindent {\underline{Case 2}}:  $i \leq t-3$.

Note that under
this assumption we have
\begin{equation} \label{biei}
b_{i+2} = e_i + 2(i+2) +1 = e_i + 2i + 5
\end{equation} and
\begin{equation} \label{binom coeff}
b_{i+2} =
\binom{i+4}{2} \hbox{ if and only if } e_i = \binom{i+2}{2}.
\end{equation}

Now note that
\begin{equation} \label{fact1}
b_{i+3} - b_{i+2} = e_{i+1} - e_i +2.
\end{equation}
 What is the $i$-binomial expansion of $e_i$?  We have
\[
e_i = \binom{m_i}{i} + \binom{m_{i-1}}{i-1} + \dots + \binom{m_j}{j}
\]
for some $m_i > m_{i-1} > \dots > m_j \geq j \geq 1.$  Since $\binom{a}{b} =
\binom{a}{a-b}$ for $a>b$, and taking into account (\ref{binom coeff}), we see
that without loss of generality we may write
\[
\begin{array}{rcl}
e_i & = & \displaystyle \left [ \binom{i+1}{1} + \binom{i}{1} +  \dots +
\binom{k}{1} \right ] + \left [ \binom{k-1}{0} + \dots + \binom{k-l}{0} \right ]
\\ \\
& = & [(i+1) + i +  \dots + k ] + l \\
\end{array}
\]
for some $k,l$.  Then from (\ref{biei}) we get
\[
b_{i+2} = [(i+3) + (i+2) + (i+1) + i + \dots + k] + l.
\]
Similarly we obtain
\[
\begin{array}{rcl}
e_{i+1} & = & [(i+2) + \dots + k'] + l' \\ \\
b_{i+3} & = & [(i+4) + (i+3) + (i+2) + \dots + k'] + l'.
\end{array}
\]
Now, we know that $\Delta H$ is an O-sequence, so the growth from $b_{i+2}$ to
$b_{i+3}$ satisfies Macaulay's bound.  That is,
\[
\begin{array}{rcl}
(i+4) + (i+3) + \dots + k' + l' & \leq & b_{i+2}^{\langle i+2 \rangle} \\
& = & [(i+4) + (i+3) + \dots + (k+1)] + l.
\end{array}
\]
Then
\[
\begin{array}{rcl}
e_{i+1} & = & b_{i+3} - (2i+7) \\ \\
& \leq & b_{i+2}^{\langle i+2 \rangle} - (2i+7) \\ \\
& = & e_i^{\langle i \rangle}
\end{array}
\]
proving that $\{ e_i : i \geq 0 \}$ is an O-sequence as claimed.  This complete
the proof of Claim~2.

\medskip

Now we have that all three sequences in (\ref{table of calculations}) are
infinite O-sequences.  Let $J$ be the lex-segment ideal in $S$ with Hilbert
function $\{ e_i : i \geq 0 \}$.  Note that since $J$ is lex-segment and $e_i =
2$ for all $i$ sufficiently large, we have that $J_i = \langle x_1,x_2
\rangle_{i}$ for all $i$ sufficiently large.

Let $Y$ be the reduced subscheme of $\mathbb P^3$ obtained by ``lifting" the
monomial ideal $J$ (cf. for instance \cite{MN2}).  $Y$ is reduced, the reduction
modulo $x_0$ of $I_Y$ is $J$, and the top dimensional part, $\bar Y$, of $Y$ is a
reduced curve of degree 2.  Note that the Betti diagram for $I_Y$ is the same as
the Betti diagram for $J$.  The first difference of the Hilbert function of
$Y$ is precisely $\{ e_i : i \geq 0 \}$.

The next claim is not needed for the rest of the proof, but is an interesting
observation.

\medskip

\noindent {\bf Claim 3:} {\em $\bar Y$ is ACM, and in fact it is a plane curve
consisting of two lines meeting in a point.}

(Note that when $d$ is odd, $\bar Y$ is just a line and there is nothing to prove
here.  But when $d$ is even, $\bar Y$ could in principle be two skew lines.  We
have to prove that regardless of the sequence $\{ e_i : i \geq 0 \}$, the curve
$\bar Y$ obtained in this way consists of two lines meeting in a point.)

The reduction of $I_{\bar Y}$ modulo $x_0$ is the ideal of a zero-dimensional
scheme of degree 2 (not necessarily saturated, a  priori, if $\bar Y$ is not ACM),
and this reduction contains $J$.  But in large degree
$J$ agrees with the ideal $\langle x_1,x_2 \rangle_{i}$, which is  the
saturated ideal of a zero-dimensional scheme of degree 2.  Therefore the
reduction of $I_{\bar Y}$ modulo $x_0$ agrees with $\langle x_1,x_2 \rangle$ in
all sufficiently large degrees, and hence its saturation is $\langle x_1,x_2
\rangle$.  But this ideal is supported at a point, so the curve
$\bar Y$ meets the hyperplane defined by
$x_0$ in  one point (up to multiplicity).  Hence $\bar Y$ is the union of two
lines meeting at this point, and we have proved Claim 3.

\medskip

We now choose polynomials $F \in (I_Y)_s$ and $Q \in R$ such that $Q$ is a
quadric and $(F,Q)$ is a regular sequence.  Let $V$ be the complete intersection
defined by $(F,Q)$.  Note that $h_V = CI$ and that $\deg V = d-2$.  We consider
the ideal
\[
I = Q \cdot I_Y + (F).
\]
The ideal $I$ is formed by a special case of Generalized Liaison Addition
\cite{GM4}.  The following can easily be deduced using \cite{GM4}.

\begin{enumerate}
\item $I$ is a saturated ideal defining a curve, $X$, of degree $d$.

\item The Hilbert function of $X$ is $H$.  (This comes directly from (\ref{table
of calculations}), using the methods of \cite{GM4}.)

\item $X$ is reduced, and as sets, we have $X = Y \cup V$.  Since $Q$ can be
chosen freely, $V$ avoids any 0-dimensional components of $Y$.

\item $F$ can be chosen to be a union of planes, as can $Q$, in which case $X$ is
a union of lines and reduced points.

\item \label{exact sequence} There is an exact sequence
\[
0 \rightarrow R(-s-2) \rightarrow I_Y(-2) \oplus R(-s) \rightarrow I \rightarrow
0.
\]
\end{enumerate}

Now we can say something about the Betti diagram of $I$.
Suppose that $I_Y$ has minimal free resolution
\[
0 \rightarrow \mathbb F_3 \rightarrow \mathbb F_2 \rightarrow \mathbb F_1
\rightarrow I_Y \rightarrow 0.
\]
Notice that because of
the values of $\{ e_i : i \geq 0 \}$ in degrees $\geq s-3$ (recall the shift in
(\ref{table of calculations})), we obtain from \cite{cho-iarrobino} that
$S/J$ (the reduction of $R/I_Y$ by $x_0$) has a non-zero socle element in degree
$s-3$ (where the ``3'' is), and in fact the socle in this degree is
1-dimensional.  Hence  since $I_Y$ and
$J$ have the same Betti diagram, we get
\[
\mathbb F_3 = R(-s) \oplus A
\]
for some free module $A$.  Because the Hilbert function of $S/J$, $(\dots, 3, 2, 2
\dots)$ (with the 3 in degree $s-3$) has maximal growth from degree $s-2$ to
degree $s-1$, the ideal $J_{\leq s-2}$ generated by the components of degrees
$\leq s-2$ has regularity $s-2$.  In fact, since the value of the Hilbert
function of $S/J$ is 2 from that point on, there are no further generators and
without loss of generality we may substitute $J$ for $J_{\leq s-2}$.  Hence we
also have that $A$ has no summands $R(-s-1)$ or higher, and that $\mathbb F_2$ has
no summand $R(-s)$ or higher.
We get the commutative diagram
\[
\begin{array}{ccccrclcccccccccccccccccccc}
&&&&& 0 \\
&&&&& \downarrow\\
&&&& [R(-s-2) \oplus A(-2)] & \oplus & 0 \\
&&&&& \downarrow \\
&& 0 && \mathbb F_2(-2) & \oplus & 0 \\
&& \downarrow &&& \downarrow \\
&& R(-s-2) & \stackrel{\alpha'}{\longrightarrow} & \mathbb F_1(-2) & \oplus &
R(-s)
\\ && \downarrow &&& \downarrow \\
0 & \rightarrow & R(-s-2) & \stackrel{\alpha}{\longrightarrow} & I_Y(-2) & \oplus
& R(-s) &
\rightarrow  & I & \rightarrow & 0 \\
&& \downarrow &&& \downarrow \\
&& 0 &&& 0
\end{array}
\]
where $\alpha = (F,Q)$.  From the associated mapping cone and the observations
above, we get the following Betti diagram for $R/I$:

\begin{equation} \label{betti diag}
\begin{array}{cccccccccccccccccccccc}
0 & : & 1 & * & * & * \\
1 & : & - & * & * & * \\
\vdots & & \\
s-2 & : & - & * & * & * \\
s-1 & : & - & * & * & 1 \\
s & : & - & - & 1 & - \\
s+1 & : & - & - & - &  -
\end{array}
\end{equation}
Note that in particular, $I$ has no minimal generators in degree $s+2$.

\begin{remark} \label{splitting}
Because of the way that the maps go in the commutative diagram, the two 1's in
the Betti diagram do not cancel (i.e.\ the corresponding terms in the free
resolution do not split).

However, if $I_Y$ had a minimal generator in degree $s$, and if we chose $F$ such
that it were such a generator, then $\alpha'$ would map $R(-s-2)$ isomorphically
onto a summand of $\mathbb F_1(-2)$ (corresponding to this generator).  In this
case, there would be a term $R(-s-2)$ in the first free module in the free
resolution of
$I$ coming from the mapping cone, and it would split off with the copy of
$R(-s-2)$ that we have described in the second free module.  Of course $I_Y$ does
not have a minimal generator of degree $s$, but in the next section we will
modify things so that such a generator does exist.  \qed
\end{remark}

Again, the case $d$ odd
is entirely analogous, and is left to the reader.  (This time we use a complete
intersection of type $(2, \frac{d-1}{2}$.)

\section{Main results}

In this section we make minor modifications on the construction of the last
section in order to get our results.

\begin{theorem} \label{first main result}
Let $H = \{ h_i : i \geq 0 \}$ be a  twice differentiable O-sequence over $R$,
with invariants $d, t, s$ as in Notation \ref{def of t}; in particular, we
continue to assume that
\[
t \leq s-1 = \left \lfloor \frac{d-3}{2} \right \rfloor.
\]
  Let $\bar H$
be the truncation at the value
$h_{s+1}$, so $\bar H$ is the Hilbert function of some set of $h_{s+1}$
points in
$\mathbb P^3$.  Then ${\mathbb B}'_{\bar H}$ does not have a unique
smallest element, and the postulation scheme  ${\mathbb H} \hbox{\rm red}^{\bar
H}(\mathbb P^3)$ is reducible.  Furthermore, on one
component,
$\mathcal H_1$, the general point corresponds to a set of points with WLP, while
on another component,
$\mathcal H_2$, no point corresponds to a set of points with WLP.
\end{theorem}

\begin{proof}
We continue to assume that $d$ is even, leaving the case $d$ odd for the reader
(but occasionally remarking on the case of $d$ odd). Notice that the first
difference of $\bar H$ is
\[
\Delta \bar H = (1,3, 6, b_3,\dots,b_{t-1},d,\dots,d, 0)
\]
where the last $d$  occurs in  degree s+1.

Consider the sequence $\{ e_i' : i
\geq 0 \}$ defined by
\[
e_i' = \left \{
\begin{array}{rll}
e_i & \hbox{if $i \leq s-1$} \\
0 & \hbox{if $i \geq s$}.
\end{array} \right.
\]
Note that $\{e_i'\}$ ends with the subsequence $(\dots, 3,2,2,0)$.  Let $J'$ be
the lexsegment ideal in $S$ with Hilbert function $\{ e_i' : i \geq 0 \}$.  Note
that $J'$ has two minimal generators in degree $s$.  Let
$Y'$ be the reduced (but now zero-dimensional) subscheme of $\mathbb P^3$ obtained
by ``lifting'' the monomial ideal $J'$.  Note that $J'$ agrees with the ideal $J$
described above in degrees $\leq s-1$, so the same is true of $I_{Y'}$ and $I_Y$,
respectively.  The first difference of the Hilbert function of $Y'$ is $\{ e_i' :
i \geq 0 \}$. Since the ideal
$I_Y$ has no generator in degree
$s$,
$I_{Y'}$ must have exactly two minimal generators in degree $s$.  (When $d$ is
odd,
$I_{Y'}$ has one minimal generator in degree $s$.)

Now we choose $F \in (I_{Y'})_s$ and $Q \in
R_2$ as above, but we choose $F$ to be a minimal generator of $I_{Y'}$.
We  form the ideal $I' = QI_{Y'} + (F)$.  This is again a saturated ideal,
defining a reduced subscheme, $X'$, of $\mathbb P^3$ which consists of the
disjoint union of the curve defined by $(F,Q)$ (which has degree $d-2$) and $Y'$.
Let $H'$ be the Hilbert function of $X'$.  We make the following Hilbert function
calculation using (\ref{table of calculations}):

 {\scriptsize
\begin{equation} \label{table of calculations 2}
\begin{array}{l|ccccccccccccccccccccccccccccccccccc}
\deg & 0 & 1 & 2 & 3 & 4 & \dots & t-1 & t & t+1 & \dots  & s-1 & s &
s+1 & s+2 & s+3 & \dots \\
\hline
e_{i-2} &&& 1 & e_1 & e_2 & \dots & e_{t-3} & e_{t-2} & e_{t-1} & \dots & 3 &
2 & 2 & 0 & 0\\
\Delta CI & 1 & 3 & 5 & 7 & 9 & \dots & 2t-1 & 2t+1 & 2t+3 & \dots  &
2s-1 & 2s &  2s & 2s & 2s &  \dots \\ \cline{1-17}
\Delta H' & 1 & 3 & 6 & b_3 & b_4 & \dots & b_{t-1} & d & d & \dots  & d & d
& d & d-2 & d-2 &  \dots
\end{array}
\end{equation}
}
Note that $I' = I_{X'}$ agrees with $I_X$ in degrees $\leq s+1$, and its Hilbert
function is $H$ up to and including degree $s+1$.  Note also that $I_{X'}$ has
only one minimal generator in degree $s+2$ (thanks to Remark \ref{splitting}),
rather than the two that one might guess by looking at (\ref{table of calculations
2}).

Thanks to
\cite{GMR}, there exists a set of points, $Z'$, on $X'$ whose Hilbert function is
the truncation of $h_{X'}$ at level $h_{s+1}$, i.e. whose Hilbert function is
$\bar H$.  The table (\ref{table of calculations 2}) shows that $I_{Z'}$ has
$d-2$ more generators in degree $s+2$ than does $I_{X'}$, so in fact $I_{Z'}$ has
$d-1$ minimal generators in degree $s+2$.

Thanks to the Betti diagram (\ref{betti diag}) and Remark
\ref{splitting}, we have the following Betti diagram for $R/I_{Z'}$:
\begin{equation} \label{betti diag for Z'}
\begin{array}{cccccccccccccccccccccc}
0 & : & 1 & * & * & * \\
1 & : & - & * & * & * \\
\vdots & & \\
s-2 & : & - & * & * & * \\
s-1 & : & - & * & * & 1 \\
s & : & - & - & - & - \\
s+1 & : & - & d-1 & * &  *
\end{array}
\end{equation}

Now we let $C$ be an ACM curve with Hilbert function $H$.  Since $\reg(I_C) =
t+1$, $R/I_C$ has Betti diagram
\begin{equation} \label{betti diag of C}
\begin{array}{cccccccccccccccccccccc}
0 & : & 1 & * & * & - \\
1 & : & - & * & * & - \\
\vdots & & \\
t-1 & : & - & * & * & - \\
t & : & - & * & * & - \\
t+1 & : & - & - & - &  -
\end{array}
\end{equation}
Let $Z$ be a finite subset of $C$ whose Hilbert function is the truncation of $H$
at level $h_{s+1}$, i.e. $h_Z = \bar H$; note that as before $I_Z$ agrees with
$I_C$ in degrees
$\leq s+1$.  Since $t \leq s-1$, the Betti diagram of $R/I_Z$ is
\begin{equation} \label{betti diag for Z}
\begin{array}{cccccccccccccccccccccccccc}
0 & : & 1 & * & * & - \\
1 & : & - & * & * & - \\
\vdots & & \\
t-1 & : & - & * & * & - \\
t & : & - & * & * & - \\
t+1 & : & - & - & - & - \\
\vdots \\
s-1 & : & - & - & - & - \\
s & : & - & - & - & - \\
s+1 & : & - & d & * &  *
\end{array}
\end{equation}
Comparing the Betti diagrams (\ref{betti diag for Z'}) and (\ref{betti diag for
Z}), one sees that $\beta'_{3,s+2} > \beta_{3,s+2}$ and $\beta_{1,s+2} >
\beta'_{1,s+2}$.  In fact it is clear that the diagrams are strongly incomparable
(in the sense of Definition \ref{strongly incomparable def}), since there can be
no element smaller than both.  The non-existence of a unique smallest element and
the reducibility of the postulation scheme then follow immediately.  $Z$
corresponds to a point in
$\mathcal H_1$ and $Z'$ corresponds to a point in $\mathcal H_2$.

The assertion about WLP comes from the observation (from the Betti diagram) that
the Artinian reduction of
$R/I_{Z'}$ has a socle element in degree $s-1$.  But the value of the Hilbert
function of this Artinian reduction in both degrees $s-1$ and $s$ is $d$, so the
map induced by a general linear form can be neither injective nor surjective
(since it has a one-dimensional kernel).  Hence this Artinian reduction does not
have WLP.  On the other hand, the Artinian reduction of $R/I_Z$ does have WLP --
 it follows from the Cohen-Macaulay property of $R/I_C$ and the fact that
$R/I_C$ agrees with $R/I_Z$ in all degrees $\leq s+1$.

Again, the case $d$ odd is almost identical and is left to the reader.
\end{proof}

\begin{corollary}
Under the assumptions of Theorem \ref{first main result}, if
$H$ is of {\em decreasing type} (i.e.  $\Delta^2 H$ is
strictly decreasing once it starts to decrease, so $H$ is the Hilbert function of
an {\em irreducible} ACM curve in $\mathbb P^3$) then the general element of
$\mathcal H_1$ also satisfies UPP.
\end{corollary}

\begin{proof}
It is clear from the proof of Theorem \ref{first main result}.  We simply choose
$C$ to be an irreducible ACM curve and $Z$ to be a general set of points on $C$
of the right cardinality.  We do not know if the general element of $\mathcal
H_2$ has UPP, although we believe that it does not.
\end{proof}

\begin{example}
Let
\[
\Delta H = (1, 3, 6, 10, 14, 16, 17, 17 , \dots).
\]
Then  $t = 6$, $d=17$ and $ s = 8$,  so Theorem \ref{first main result} considers
the Hilbert function
\[
\Delta \bar H = (1, 3, 6, 10, 14, 16, 17, 17, 17, 17,0).
\]
That is,
\[
\bar H = (1, 4, 10, 20, 34, 50, 67, 84, 101, 118, 118, \dots).
\]
The corresponding postulation scheme $\mathbb H \hbox{ilb}^{\bar H}(\mathbb P^n)$,
parameterizing sets of 118 points in
$\mathbb P^3$ with Hilbert function $\bar H$, is reducible.  \qed
\end{example}

\begin{corollary} \label{main corollary}
Let $H = \{ h_i : i \geq 0 \}$ be a twice differentiable O-sequence, with
invariants $d,t,s$ as in Notation \ref{def of t}; in particular, we assume that
\[
t \leq s-1 = \left \lfloor \frac{d-3}{2} \right \rfloor.
\]
Write $\Delta H = \{ 1, 3, b_2, b_3, \dots, b_{t-1}, d, d, \dots \}.$
Let $\bar H$ be the Hilbert function whose first difference is
\[
\Delta \bar H = \{ 1, 3, b_2, b_3, \dots, b_{t-1}, d, \dots, d, b_{s+2}, b_{s+3},
\dots , b_r, 0 \}
\]
where
\[
\left \{
\begin{array}{ll}
b_{s+2} \leq d-2 & \hbox{if $d$ is even} \\
b_{s+2} \leq d-1 & \hbox{if $d$ is odd}
\end{array}
\right.
\]
 and $b_i \geq b_{i+1}$ for all $i \geq
s+2$.  Then all the conclusions of Theorem \ref{first main result} continue to
hold for $\bar H$.
\end{corollary}

\begin{proof}
As before, we prove the case where $d$ is even, leaving to the reader the case of
$d$ odd. Our approach here is similar to that taken in \cite{HMNW}, Proposition
3.5.

We return to the monomial ideal $J'$ and the  reduced subscheme
$X'$ obtained at the beginning of the proof of Theorem \ref{first main result}.
By the mechanism of liftings of monomial ideals (cf.\ \cite{MN2}), we may assume
that $F \in (I_{Y'})_s$ is a union of planes.  Clearly $Q$ can also be chosen to
be a (general) union of planes.  Hence $X'$ is the disjoint union of $2s = d-2$
lines (defined by the complete intersection of $F$ and $Q$) and a finite set of
points ($Y'$) whose regularity is $s$.  We saw that the Hilbert function of $X'$
has first difference
\begin{equation} \label{delta h of X'}
(1,3,6,b_3,\dots,b_{t-1}, d,\dots,d, d-2, d-2,\dots)
\end{equation}
with the last $d$ occurring in degree $s+1$.

Note that the union, $A$, of $2s = d-2$ lines is a complete intersection of type
$(2,s)$; in particular it is ACM.  Its Hilbert function has first difference
\[
(1,3,5, \dots, d-5, d-3, d-2, d-2 \dots )
\]
where the first $d-2$ occurs in degree $s$.
It is easy to check (e.g.\ by considering liftings) that we may order the lines
$\lambda_1,\lambda_2,\dots , \lambda_{d-2}$ such that for each $i$, $A_i
:= \lambda_1
\cup \dots \cup \lambda_i$ is ACM.  Notice that the Hilbert function of $A_i$ has
first difference which reaches value $i$ in degree $< s$, except for $A_{d-2}$,
which reaches value $d-2$ in degree exactly $s$.

For any $i$, the base locus of the linear system $|(I_{X'})_i|$ includes all the
lines $\lambda_j$.    Furthermore, in degrees $\geq s+2$, the reduction of
$R/I_{X'}$ modulo $x_0$ agrees with the reduction of $R/(F,Q)$ modulo $x_0$.

We will prodeed inductively.
Consider the Hilbert function
\[
1,3,b_2,\dots,b_{t-1}, d, \dots ,d,b_{s+2}, b_{s+2}, \dots.
\]
This can be obtained by the ideal
\[
I_{X_{s+2}} := I_{Z'} \cap I_{\lambda_1} \cap \dots \cap I_{\lambda_{b_{s+2}}}
\]
i.e. by taking the ideal of $X_{s+2} := Z' \cup A_{b_{s+2}}$, where $Z'$ is the
reduced zero-dimensional scheme obtained in Theorem \ref{first main result}.
Indeed, since $\lambda_1,\dots,\lambda_{b_{s+2}}$ are contained in the base locus
of
$I_{Z'}$ in degrees $\leq s+1$, they impose no additional conditions in those
degrees.  And in degrees $\geq s+2$ the reduction modulo $x_0$ sees only the
$b_{s+2}$ lines.  We choose a finite set of points $Z_{s+2}$ whose Hilbert
function is the truncation, i.e. whose Hilbert function has first difference
\[
1,3,b_2,\dots,b_{t-1}, d, \dots ,d,b_{s+2}, 0.
\]

For the next step we proceed as we have done here, but substituting $X_{s+2}$ for
$X'$ and $Z_{s+2}$ for $Z'$, and we produce in the end a finite set of points
with Hilbert function having first difference
\[
1,3,b_2,\dots,b_{t-1}, d, \dots ,d,b_{s+2}, b_{s+3}, 0.
\]
After a finite number of steps, we are finished.  Let us denote by $W'$ this
reduced zero-dimensional scheme with the desired Hilbert function.

For the scheme $W$ (analogous to $Z$ in Theorem \ref{first main result}) having
the same Hilbert function as $W'$ but strongly incomparable Betti diagram, we
first note that a curve $C$ can be constructed with Hilbert function having first
difference
$\{ 1, 3, b_2, \dots, b_{t-1}, d, d, \dots \}$ and such that $C$ is ACM, and as
before, it is a union of lines $\lambda_1 \cup \lambda_2 \cup \dots \cup
\lambda_d$ such that $\lambda_1 \cup \dots \cup \lambda_i$ is ACM for all $i$.
We then proceed exactly as we did above, considering truncations and slowly
removing lines, to produce a reduced zero-dimensional scheme with the desired
Hilbert function.

Now we have to check that these zero-dimensional schemes $W$ and $W'$ have
strongly  incomparable Betti diagrams.  Note that the Betti diagram for $R/I_C$ is
the same as that in (\ref{betti diag of C}), by construction.

Recall that in Theorem \ref{first main result}, we
produced zero-dimensional schemes $Z$ and $Z'$ with Hilbert function having first
difference
\[
1,3,b_2,\dots,b_{t-1}, d, \dots ,d,0
\]
where the last $d$ occurs in degree $s+1$.  We saw that the Betti diagram for
$R/I_{Z'}$ was
\[
\begin{array}{cccccccccccccccccccccc}
0 & : & 1 & * & * & * \\
1 & : & - & * & * & * \\
\vdots & & \\
s-2 & : & - & * & * & * \\
s-1 & : & - & * & * & 1 \\
s & : & - & - & - & - \\
s+1 & : & - & d-1 & * &  *
\end{array}
\]
while the Betti diagram for $R/I_Z$ was
\[
\begin{array}{cccccccccccccccccccccccccc}
0 & : & 1 & * & * & - \\
1 & : & - & * & * & - \\
\vdots & & \\
t-1 & : & - & * & * & - \\
t & : & - & * & * & - \\
t+1 & : & - & - & - & - \\
\vdots \\
s & : & - & - & - & - \\
s+1 & : & - & d & * &  *
\end{array}
\]
Now instead we have constructed reduced zero-dimensional schemes $W$ and $W'$
with Hilbert function having first difference
\[
1, 3, b_2,\dots,b_{t-1}, d,\dots,d,b_{s+2}, \dots.
\]
In particular, $I_{Z'}$ (resp.\ $I_{Z}$) agrees with $I_{W'}$ (resp.\ $I_{W}$) in
all degrees $\leq s+1$.  It follows that the Betti diagram for $R/I_{Z'}$ (resp.\
$R/I_{Z}$) agrees with the Betti diagraom for $R/I_{W'}$ (resp.\ $R/I_{W}$) in
all  rows above the one labelled $s$.  The fact that the Hilbert function in
degree $s+2$ is now $b_{s+2}$ rather than 0 means that instead of having $d-1$ and
$d$ minimal generators in degree $s+2$, respectively (in the above two diagrams),
we instead have $d-1-b_{s+2}$ and $d-b_{s+2}$, respectively.  But the
incomparability of the diagrams is preserved.
\end{proof}

\begin{example}
In Example 3.4 of \cite{geomhilb} it was shown that if $\bar H$ is the Hilbert
function with first difference $(1,3,6,9,11,11,11,0)$ (corresponding to a set of
52 points in
$\mathbb P^3$), then $\mathbb
H\hbox{\rm red}^{\bar H}(\mathbb P^3)$ is reducible.  Since here $d=11, t=4$ and
$s = 5$, this is easily seen to follow immediately from Theorem \ref{first main
result}.  Furthermore, by Corollary \ref{main corollary}, also the Hilbert
function $H'$ with first difference
\[
\begin{array}{c}
(1,3,6,9,11,11,11, 9,6,3,1) \ \hbox{ or } \\
(1,3,6,9,11,11,11, 10, 8, 8, 5, 5, 5, 4, 3,3, 1)
\end{array}
\]
 has the property that $\mathbb
H\hbox{\rm red}^{H'}(\mathbb P^3)$ is reducible.

\end{example}

\begin{example} \label{ci ex}
Richert shows that even when the Hilbert function is one of a complete
intersection, the conclusions about not having a unique minimum element of
$\mathbb B_H$ (or about the reducibility of $\hredn$) may hold.  His infinite
family of examples deals with the Hilbert function of a complete intersection of
type $(m,m+1,2m+1)$.  More generally, in
\cite{RZ2} Ragusa and Zappal\`a show that
if $H$ is the Hilbert function of a complete intersection of type $(a,b,c)$ with
$a \leq b \leq c$ and $b+3 \leq c \leq ab$, and if $(a,b,c) \neq (4,4,7)$, then
$\mathbb B_H$ does not have a unique minimum element.  Using our methods it is
also possible to obtain the Hilbert function of a complete intersection, but the
``flat part" of the Hilbert function becomes rather large.  For example, taking
$a=b=7$ requires
$c=26$, and we obtain the Hilbert function
{\small
\[
\begin{array}{lcccc}
(1,3,6,10,15,21,
28,34,39,43,46,48,49,49,49,49,49,49,49,49,49,49,
49,49,49,49, 48, 46, 43, \\ \\  39, 34,
28, 21, 15, 10, 6, 3, 1).
\end{array}
\]}

One might wonder if we could improve the method using a different kind of
complete intersection $CI$ in our construction (recall that we used one of type
$(2,
\lfloor \frac{d-1}{2} \rfloor)$).  The idea would be to try to diminish the
number of copies of $d$ in the first difference of the Hilbert function.  If our
goal were some restricted class of Hilbert functions, such as those of complete
intersections of type
$(a,b,c)$, experimental evidence suggests that it might be possible.  But since
this case is known \cite{RZ2}, we have to explore the general case.

Suppose, for instance, that we wanted to show that the Hilbert function with first
difference
\[
\Delta H = (1,3,6,10, 15, 19, 23, 26, 27, 28, 29, 29, \dots, 0)
\]
corresponds to a postulation scheme $\hredthree$ which is
reducible.  Using the method of this paper, we use a complete intersection of type
$(2, 14)$ and study the table

 {\small
\begin{equation} \label{ci2}
\begin{array}{l|ccccccccccccccccccccccccccccccccccc}
\deg & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 & 15 & \dots
\\
\hline
\Delta H & 1 & 3 & 6 & 10 &15 & 19 & 23 & 26 & 27 & 28 & 29 & 29 & 29 & 29 &
29 & 29 & \dots
\\
\Delta CI & 1 & 3 & 5 & 7 & 9 & 11 & 13 & 15 & 17 & 19 & 21 & 23 & 25 & 27 & 28 &
28 &
\dots
\\
\cline{2-18} &&& 1 & 3 & 6 & 8 & 10 & 11 & 10 & 9 & 8 & 6 & 4 & 2 & 1 & 1 &
\dots
\end{array}
\end{equation}
}
Here $t = 10$ and $s = 14$.  Note that the bottom row is indeed an O-sequence.
Theorem \ref{first main result} then draws a conclusion about the Hilbert
function with first difference
\[
(1,3,6,10,15,
19,23,26,27,28,29,29,29,29,29,29,0).
\]

But if we tried to reduce
the number of 29's by choosing, for instance, a complete intersection of type
$(4,7)$, we obtain the diagram
{\small
\[
\begin{array}{l|ccccccccccccccccccccccccccccccccccc}
\deg & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 &   \dots
\\
\hline
\Delta H & 1 & 3 & 6 & 10 &15 & 19 & 23 & 26 & 27 & 28 & 29 & 29 &  \dots
\\
\Delta CI & 1 & 3 & 6 & 10 & 14 & 18 & 22 & 25 & 27 & 28 & 28 & 28 &
\dots
\\
\cline{2-14} &&&&& 1 & 1 & 1 & 1 & 0 & 1 & 1 & 1 &
\dots
\end{array}
\]
}
and the bottom line is not an O-sequence.

However, in (\ref{ci2}) if we instead use a complete
intersection of type $(3,9)$ then we obtain
 {\small
\[
\begin{array}{l|ccccccccccccccccccccccccccccccccccc}
\deg & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 &  \dots
\\
\hline
\Delta H & 1 & 3 & 6 & 10 &15 & 19 & 23 & 26 & 27 & 28 & 29 &
29 & 29 & \dots
\\
\Delta CI & 1 & 3 & 6 & 9 & 12 & 15 & 18 & 21 & 24 & 26 & 27 & 27 & 27 & \dots
\\
\cline{2-15} &&&& 1 & 3 & 4 & 5 & 5 & 3 & 2 & 2 & 2 & 2 &
\dots
\end{array}
\]
}
One might think that the method of this paper would allow us to start with the
O-sequence $(1,3,4,5,5,3,2,2,0)$ and proceed as before to obtain a result about
the Hilbert function with first difference $(1,3,6,10, 15, 19, 23, 26, 27, 28,
29,0)$.  But recall that the complete intersection needs to include a minimal
generator for $I_{Y'}$, and the O-sequence $(1,3,4,5,5,3,2,2,0)$ does not allow a
minimal generator of degree 9.  So in fact we have to start with the O-sequence
$(1,3,4,5,5,3,2,2,2,0)$ and then we can indeed draw a conclusion about the Hilbert
function with first difference {\small $(1,3,6,10,15,
19,23,26,27,28,29,29,0).$ } This is not covered by Theorem \ref{first main
result}.

Another obstruction to extending this technique is that if $CI$ is the Hilbert
function of a complete intersection of type $(a,b)$, then in order for the
sequence  $\{ e_i \}$ to begin with a 1 we need  $a <
b$, and also we need that $a$ be strictly smaller than the initial degree of
$H$ (or some small variant which we need not make explicit here).  If we impose
this condition, and also the condition that
$a+b \geq t+3$ and that $ab$ is smaller than $d$ but ``not too much smaller" (so
that we have maximal growth of the first difference of the Hilbert function) then
very likely the technique could be extended, as illustrated above.  But the gain
would only be to shorten the string of $d$'s in the middle of the
$h$-vector.  Given the complicated
nature of these numerical conditions (we have not even addressed what we
would need to ensure that $\{ e_i : i \geq 0 \}$ is an O-sequence), and the fact
that already our result gives a very large new class of Hilbert functions for
which the postulation scheme is reducible, it does not seem worthwhile to pursue
this.
\qed
\end{example}

redpajama_set_name: RedPajamaArXiv


Abstract The Ultra-wide-band (UWB) radio is an emerging data rate radio technology intended for low transmission power wireless multimedia applications. The significant advantages of this technology are the low power operation, mitigated multi-path fading effects and unique precise position/timing location ability. In a pulse-based UWB system, very short pulses are modulated and transmitted at a very low power level. Hence, it is important to develop the low power UWB Transceiver. The project presents the design and implementation of a single chip CMOS Low Power UWB Transceiver for short range communications. In the UWB transmitter, the most important point is to generate the short pulses in a well-defined bandwidth. A new Gaussian pulse filter is designed to produce an approximated Gaussian pulse train which has a reduced side-lobe frequency spectrum. The two key points in the design of the new transmitter are the low power and the small chip size. The first transmitter was designed for multi-band operations over the frequency range from 3GHz to 5GHz using the Pulse-Amplitude Modulation (PAM) scheme. The design is based on a 0.18-µm CMOS Baseline process. The core layout size is less than 0.2 mm2. The simulation results show that the generated signals satisfy the FCC spectrum mask. The average power consumption is measured 1.97mW at a 1.8 V supply voltage. After the optimization, the power consumption of the transmitter can be reduced to as low as 1.08mW. Subsequently, the transmitter circuit has been modified to enable two selectable modulation schemes PAM and Pulse-Position Modulation (PPM). The average power consumption is less than 2mW for both types of signaling at a 1.8 V supply voltage. Pulses are transmitted at a PRF (Pulse Repetition Frequency) of 52MHz in multiple 520 MHz bandwidth channels equally spaced within the 3-5 GHz UWB lower band. For the receiver, the Low Noise Amplifier (LNA) has been designed for the same frequency band from 3GHz to 5GHz with a variable gain controlled by tuning the bias voltage. After the amplification, the received signal is down converted to the baseband by the quadrature passive mixers where the quadrature local oscillators are implemented using the Series Quadrature VCO (S-QVCO). Simple low pass filters (LPF) are used to filter the high frequency noise at the mixer outputs. In order to synchronize the pulse polarities, the CMOS full wave rectifiers have been used to detect the pulses in I and Q paths which are then combined to recover the detected baseband pulses. The proposed transmitter and receiver have been integrated into a new single chip UWB transceiver. The total die size of the chip is 3.1 mm2. The performance of the transmitter is similar as above. At the gain of 39dB, the receiver consumes 17mA for the 1.8 V supply voltage. The total power consumption of the transceiver is 30.6 mW. The operating frequency of this transceiver is from 3GHz to 5GHz over multiple 500 MHz bands. Overall, this research work focuses on a low power UWB transceiver with small chip area. Theoretical analysis, simulation results and measurement results are presented to verify the analysis and effectiveness of the proposed design. This work has also been published in technical journals and conferences.


Fast 'bling' cars usually go hand in hand with the catch phrase 'toys for boys'; they're big, and bulky, go faster than the speed of sound and women are usually completely intimidated at the thought of even driving one. Well ladies, times they are changing. When Rave Review! was given the opportunity to test-drive Nissan's new 370Z Roadster we excitedly threw our hands up in the air and relished the challenge that awaited us. We were going to be driving in style.
The first thing that got my attention as the Nissan 370Z pulled up the driveway was 'that's one good looking car'. With 20" RAYS wheels holding it up, its design is smooth, and balanced. It catches your attention right away. Not to mention the eyes of every driver we passed on the road. The design is a sight, will make you weak in the knees and tells you straight out 'I am a Sportscar' and believe me when I tell you, it looks the part.
The cars interior is just as attractive as the outside. The aluminum blended with leather is one classy touch making you feel like a Hollywood starlet. Every button you'd ever need to reach in the car is at arms length. The seats are so comfortable; you almost sink into them like it was especially made for you to sit in. Another cool feature is that the seats can be warmed up or cooled down, depending on how you feel. Your steering wheel is covered in leather, is a pleasure to hold and is adorned with buttons to control your mobile phone, On Board Navigation, Audio and Cruise control. The inside seats just two people, but there was plenty of space and nobody felt claustrophobic. There's even a little space behind you to throw your gym bag, or handbag. The boot is not very big at all, but thats usually the case with a Roadster. You will find thought that there is enough space for a quick shop, or your golf clubs.
Now we must admit, we had a little fun bringing the top up and down…as girls would. It's so easy and works at the push of a button. The beautifully lined cover lifts up and folds down in an instant. Once we had the top down it was time to see what this baby could do. And it did not disappoint. With 332-horse power this car soars and you don't have to rev the car at all to get it going to ridiculous speeds. Pulling away is a breeze and when it comes time to break, you never get the feeling that it wont stop, no matter how fast you're going.
This is one solid car, as solid as a Tiffany silver-plated ring ladies. We never felt one squeak or uncomfy movement. Best thing about the Nissan 370Z is that the base car comes fully loaded with every single extra imaginable. That's right, you don't pay extra for one thing. Now we don't know any other brand of Sportscar that can boast that.
Whats the moral of the story girls? We can drive any fast car just as well as the boys can, and when we need to get to a 50% off sale in record time; we just may drive it better!
**Rave Review! would love to thank Nissan South Africa for the opportunity to test drive this amazing car.
***The Nissan 370Z is available from all Nissan Dealerships in South Africa. Visit them at www.370z.co.za or www.nissan.co.za for more info.


text:
Marco Borsato confirms a surprise performance at the Pinkpop festival this year
Marco Borsato will give on Sunday is, indeed, a surprise performance at the Pinkpop festival this year. That is, the singer herself confirmed from the date of notification of The Reply.
"And here I thought that it was a secret", send a tweet, Staged Sunday morning.
'Marco, He takes care of Sunday's surprise performance at the Pinkpop festival'
And I have to say that it was a secret 😂😂😂 https://t.co/Bnsade7srt
AvatarAuteurborsatoMoment of plaatsen09:10 – June 9, 2019
Later, he added on Instagram with the following comments: "It was really a surprise, but yes, I can confirm that we are on tonight as a guest with Armin van Buuren at the Pinkpop festival this year. My first time to the Pinkpop festival this year, along with Davina, Michelle, John, Ewbank, our single, How to dance, do so. Man, man, I was looking forward to it."
The first one is called close on Sunday, the second day of the festival. It has already been announced that Van Halen singer David Lee Roth, and the dj herself.
The 52-year-old so He was never at the Pinkpop festival this year. The singer recently gave five concerts in The Kuip in Rotterdam, with Davina Michelle, one of the guests was made.
50 years of the Pinkpop festival this year: "National and international acts broke through in Glasgow'
Jennifer Lopez sings duet with his daughter for the first show of the new tour
The Pinkpop festival is calling for visitors in the future to arrive, due to predicted bad weather
The Concert of Jennifer Lopez will be truncated in New York city due to power outage
The concert of Jennifer Lopez at the world famous Madison Square Garden on Saturday night (local time) in New York for an early stop due to a major power outage in the city. Also, the musicals on Broadway have been cancelled. The traffic lights in...
Recensieoverzicht new album of Ed Sheeran: 'rockmodus you want him not to be'
Singer and songwriter, Ed Sheeran, launched on Friday its new album, " No." 6 Collaborations Project, which includes collaborations can be heard on his, Justin Bieber, Cardi B, and Eminem. What is the Dutch music critics for the album? A...
New music Fridays: a single Performed in The Lion King
Each and every Friday it is refreshed with the muziekstreamingdienst Spotify playlist of New Music to Play, including a many of the new singles by national and international artists, musicians. NU.nl put in a number of high profile releases in a row...
Orders, supports twin and ASAP Rocky, said to occur in Sweden is finished
Rapper Tyga has to be located jointly in accordance with the vakgenoot ASAP Rocky, who is in custody sits in a jail in Stockholm, sweden. Orders made on Thursday announced that he will be performing in Sweden this coming Sunday claim. ASAP Rocky was...
Bon Iver at the end of August with a new album
Bon Iver brings, at the end of August with a new album, called i, ii. I, and i is the fourth full length album, the band's singer Justin Vernon. The album is partly located in the U.s. state of Wisconsin, contained, and partly in dallas, Texas...
Miley Cyrus wants of men are not pleased with the sexual performances of
Miley Cyrus is not looking for men who are happy with the sexual performance. The singer wants the men to know that they have choices in her career is not for them, she said in an interview with Elle. "I enjoy the feeling that is sexual in...
ASAP Rocky's cancellation of his shows in July due to detention in Sweden
ASAP Rocky has forced all of his appearances in the month of July and need to cancel, please notify TMZ. The assault suspect's release may be the cell that is in Sweden, not to leave. In the thirty years ' rapper, whose real name is...
The Opposites one-off return to the show on the festival, Visit
The rapduo The Opposites is time for a show at the festival, Visit on the 10th of August. It was the first and for the time being the only time that the Big2, and William perform together, since they are, in 2014, they stopped by The Opposites...
Phil says, four European shows due to illness
Phil did four shows in Belgium and Germany to cancel their appearance due to illness. Reports that the British singer himself, in a message on Twitter. Sting had a concert in Gent last Monday's cancellations and coming shows-in Munich...
House Intelligence chief and straining the social media companies in deepfake policy
The woman appears to be holding up a beer during a perverted sex act on a busy beach, 'Bloody disgusting'
12 Weird and Wonderful Facts About Lipstick
Hartapparaat with bad battery also in the Netherlands
"Possible new drug against severe asthma-attacks'
LUMC start test with antibodies against type 1 diabetes
'Dutch people know too little about deafness'
Dissatisfaction among patients about the conversation with the doctor
A third men USA dies of cancer by smoking
'Shortage of reumamedicijn sulfasalazine threatens'
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Trump win fuels donations, IUD ask at Planned Parenthood

redpajama_set_name: RedPajamaCommonCrawl


text:
Q: How to include additional directories when configuring makefiles I'm trying to compile geany-plugins-1.28. The debugger plugin (the only one I need) gives the error:
debug.c:53:21: fatal error: vte/vte.h: No such file or directory
#include <vte/vte.h>

Clearly it needs to know where this file is located to compile. I found the vte.h file in the src directory of the main program geany-1.28. When running
sudo ./configure cflags=-I/home/pi/Desktop/geany-1.28/src

I get the same error about the missing header later trying to compile the debugger plugin.
I ran
./configure --help

to get all the flag options. The output is here
How do I get this to configure correctly so that it compiles. I need to compile the debugger version 1.28 myself because apt only installs 1.24 which I think has a bug because it crashes when I run my code with the error:
close failed in file object destructor:
sys.execpthook is missing
lost sys.stderr

A: CFLAGS is case-sensitive environment variable, so you should set it before running configure, not try to pass it as a command line argument. This variant:
$ export CFLAGS=-I/home/pi/Desktop/geany-1.28/src
$ ./configure

leaves CFLAGS set for current shell until you leave it. While this:
$ CFLAGS=-I/home/pi/Desktop/geany-1.28/src ./configure

sets variable only for current command, i.e. configure.
Some other issues:

*

*You do not need sudo to configure and make. It is also unnecessary for make install if you set PREFIX to a path you have privileges to write to.

*Does plugin's build system also builds all it's dependencies? If not, you may face linker errors a bit later.

Update:
I have tried to build debugger plugin and got rid of your error. It seems that vte.h coming with Geany is it's intrinsic, while the plugin requires full-featured file from the library. So I just installed vte and vte-devel from repos. Nevertheless, I got some other unrelated errors coming from Glib. I will not continue my attempts to build all this right now. Hope my effort will be helpful at least a little.

A: As in this answer stated, vte.h is not the file you are looking for. Install libvte(-dev) package on your system and rerun configure.
Just for the record: vte.h on Geany is a dummy to allow Geany to kind of dynamical enable vte or disable it depending on vte is installed on the system or not.

redpajama_set_name: RedPajamaStackExchange


text:
Q: How should I understand "from a slightly premature but inevitable extinction" in the paragraph?
Passion seems appropriate to this discussion, because we are living at
a historic  moment, when the young are threatened with paying an
appalling price to preserve  some hundreds of thousands of us oldies,
overwhelmingly the likeliest victims of  coronavirus, from a slightly
premature but inevitable extinction. One of my wife's favourite
observations is that "none of us is going to get out of this alive",
and never has this seemed more obvious.

I read the above paragraph in this article. What I'm not sure is the meaning of 'from a slightly premature but inevitable extinction'. My understanding of this is that the young are trying hard to protect the oldies from premature death. Everyone, however, is going to leave this world, which is inevitable.
Is my understanding right? Thank you in advance.

A: You are correct.
The author is also (perhaps jokingly) saying that the death would be "slightly" premature, implying that that older persons are usually closer to death in general.

A:
when the young are threatened with paying an appalling price to preserve some hundreds of thousands of us oldies, overwhelmingly the likeliest victims of coronavirus, from a slightly premature but inevitable extinction.

This means

when the young may have to suffer an economic collapse so that old people won't have to die earlier than they would naturally, if the virus didn't exist.

The specific phrase

from a slightly premature but inevitable extinction

means that old people will die (inevitable extinction), and if they die by the virus, that death will be sooner than if it happened by other natural causes (slightly premature).
The paragraph isn't saying that the young are trying hard to keep the old people from dying, but that society is trying to keep the old from dying, and that is costing the young, in terms of the economic collapse.


Dental extraction is a type of surgery in dentistry. The main goal of dental treatment is the maintenance of the teeth health. But, unfortunately, sometimes a bad tooth cannot be restored, and then it has to be removed in order to prevent subsequent complications and return the patient to a comfortable lifestyle. After the extraction, prosthetics can be performed to restore the aesthetics of a smile and the full function of the chewing and speech apparatus.
in case of tooth alignment, as recommended by the orthodontist.
The procedure of dental extraction lasts up to 20 minutes, but in difficult cases, it may take about 40 minutes (when the tooth is horizontal or oblique or hidden under the gum). Due to effective anesthesia, the extraction is performed in a comfortable environment for patients, painlessly and easily. If necessary, this procedure can be carried out during medication sleep (Sedation Dentistry). After the removal of the tooth, the process of healing the alveolar socket begins. If you follow the recommendations of the dentist on oral hygiene during this period, then after 1.5 – 2.5 weeks the healing process will be successfully completed. Modern methods of plasma therapy significantly accelerate this process and are used in the treatment of alveolitis, if the alveolar socket is inflamed.
7-10 days after the extraction, the examination and removal of sutures are performed.


An electronic book includes a housing defining a card slot on a first face thereof, the housing including first and second housing portions; a spine member pivotally coupling the first and second housing portions, the spine defining a cavity therein; a screen display mounted on the housing on a second face thereof opposite the first face; microprocessor circuitry positioned in the housing, in between the card slot and the screen display, the microprocessor circuitry controlling an operation of the electronic book; and a card scanner mounted in the housing between the microprocessor circuitry and the card slot, and facing away from the screen display, the card scanner configured to scan a card inserted into the card slot and to convert a two-dimensional pattern on the card into data signals, the card scanner being connected to the microprocessor circuitry. The microprocessor circuitry is configured to convert the data signals into image data to be viewed on the screen display.
This application is a continuation of application Ser. No. 11/329,039 filed Jan. 11, 2006, which is a continuation application of U.S. application Ser. No. 09/942,602 filed on Aug. 31, 2001 (now abandoned), all of which are herein incorporated by reference.
The present invention relates to an electronic book or "e-book" being a device that presents text and/or graphics, for example the text of a book or magazine and associated pictures, upon an electronic screen. Such devices typically comprise a display screen, for example an LCD screen under control of a programmed microprocessor. The microprocessor reads data from a data storage medium such as a Micro-CD-ROM or memory card such as a PCMIA card and converts the data into text and/or graphics that are displayed on the LCD screen.
One commercially available electronic book is the REB1100 available from RCA. That device has a monochrome LCD touch screen and a built in 33.6 kbps v.34 capable modem that allows digital book data to be downloaded from a remote database into an onboard 8 MB memory.
In U.S. Pat. No. 6,229,502 there is described an electronic book which is configured to read digital book data from a ROM such as a PCMIA card.
In U.S. Pat. No. 6,037,954 to McMahon there is described an electronic book which includes a Micro-CD-ROM drive for reading digital book data encoded onto a Micro-CD-ROM.
One problem with these devices is that they rely on data storage or distribution systems which are relatively expensive and complex to implement.
According to one aspect of the present disclosure, an electronic book includes a housing defining a card slot on a first face thereof, the housing including first and second housing portions; a spine member pivotally coupling the first and second housing portions, the spine defining a cavity therein; a screen display mounted on the housing on a second face thereof opposite the first face; microprocessor circuitry positioned in the housing, in between the card slot and the screen display, the microprocessor circuitry controlling an operation of the electronic book; and a card scanner mounted in the housing between the microprocessor circuitry and the card slot, and facing away from the screen display, the card scanner configured to scan a card inserted into the card slot and to convert a two-dimensional pattern on the card into data signals, the card scanner being connected to the microprocessor circuitry. The microprocessor circuitry is configured to convert the data signals into image data to be viewed on the screen display.
FIG. 1 is a first perspective view of an apparatus according to a preferred embodiment of the present invention.
FIG. 2 is a second perspective view of the apparatus.
FIG. 3 is a third perspective view of the apparatus.
FIG. 4 is a perspective view of the apparatus shown open for use.
FIG. 5 is an exploded perspective view of the apparatus.
FIG. 6 is a system block diagram of the apparatus.
FIG. 7 is a cross sectional view of the apparatus open and through line B-B′ of FIG. 4.
FIG. 8 is a cross sectional view of the apparatus closed and through line B-B′ of FIG. 4.
FIG. 9 is a cross sectional view of the apparatus through line A-A′ of FIG. 4.
The drawings illustrate an electronic book that is configured to read data encoded as a pattern printed on a sheet of card.
With reference to FIGS. 1 and 2, there is depicted a view of the front of an electronic book or "e-book" 2 according to a preferred embodiment of the invention. The e-book has a foldable housing including first and second housing portions in the form of front door 6 and a rear door 8 each pivotally connected to a spine 16. A clasp 14 holds the two doors closed when the e-book is not being used. The outside of the front door 6 features a clear window 10 through which a data card 18 is visible. The data card is inserted under the window through a card slot 24 and is engaged by a roller and fed into an internal cartridge 38 (FIG. 5). On one side of the data card there is printed information for a user to read such as the title and author of a book. Accordingly a user of the e-book is able to determine at a glance the content that the e-book is loaded with. The text of the book is encoded as a pattern on the reverse side of the data card.
At the top of the outside of front door 6 there is located an eject button 12. Upon operation of the eject button, card 18 is ejected from the e-book by the internal roller mechanism.
At the base of spine 16 there is located a battery cover 4 that covers a battery compartment for accommodating two AAA size batteries that power the e-book.
The outside of rear door 8 is visible in FIG. 3. Storage magazine 20 is hinged to swing out from rear door 8 to a position, as shown, where data cards 22 may be stored or selected for removal and insertion into card slot 24.
FIG. 4 shows the e-book with the front and rear doors swung about spine 16 to an open position. In that position a flexible LCD screen 24 is visible. It is preferred that a VGA resolution monochrome screen be used being a passive bi-stable reflective polymer doped liquid crystal (PDLC) display fabricated on a flexible polymer substrate.
By using a bi-stable screen power consumption is reduced as the screen draws zero current while presenting a static image.
The LCD screen operatively displays the text of the book encoded on card 24. A user of the e-book is able to control which page of text is presented by means of joystick 26.
The internal arrangement of the e-book may be comprehended by referring to FIG. 5 which is an exploded view. It will be noted that on the underside of LCD 24 there are located two PCBs 26 and 28. PCB 28 has mounted directly upon it a scanner head 30. The PCBs 26 and 28 are loaded with various electronic components including a microprocessor, RAM and ROM memory chips and power supply conditioning circuitry. It is envisaged that a VLIW microprocessor and accompanying circuitry, as described in U.S. patent application Ser. No. 09/113,053 and hereby incorporated by reference in its entirety, be used. PCBs 26 and 28 communicate by means of conductive traces on the back of flexible LCD 24. The conductive traces terminate in peripheral contact regions 58 and 60 of the LCD screen which are folded over the edges of the PCB's to form connections with contact pads on the PCBs.
Adjacent scan head 30 there is located a motor 32 which drives roller 34 via reduction gearing. A switch 36 is provided to detect depression of eject button 12. FIG. 6 provides a further exploded view internal cartridge 38 and window 10.
Power for the electric motor and various circuit modules is conveyed from a battery compartment in the spine of the e-book to PCB 28 by means of cable 29.
A block diagram of various electronic components of the e-book is shown in FIG. 6. Power from batteries 40 is conditioned and distributed by power supply circuit 42 to the various circuit modules located on the PCBs. To extend battery life, the processor circuitry is powered down whenever the screen display is constant. Near zero power consumption allows the e-book to appear to always be "on" in the manner of a conventional paper based book.
Processing module 44 includes a central processing unit 46, which communicates with BIOS memory chip 48 and RAM 50 in the conventional manner. The CPU operates according to a program stored in program memory chip 52. The processing module receives data and control signals from eject sensor 36, joystick 26 and scanner 30. In a further, more complex implementation, LCD screen 24 may be touch sensitive in which case the processing module would also be responsive to command signals generated by a user touching the LCD screen.
In operation a book data card is inserted through card slot 24. In response card insertion sensor 48 generates a signal alerting processing module 44 to activate electric motor 32 thereby causing roller 34 to draw the card into internal cartridge 38. As the card is drawn in scan head 30 converts a pattern on the card into corresponding data signals which are decoded by CPU 46 according to an algorithm implemented in the software stored in program memory chip 52. The resulting decoded text file is stored in RAM 50.
The decoded signals are displayed as readable text on LCD 24 under control of display controller 44. Of course, as referred to previously, in magazines and some books, such as childrens' books, technical volumes and manuals, illustrations or graphics may feature prominently. Accordingly, the software stored in program memory chip 52 may also include instructions to decode figures encoded on the book data card.
The processing module 44 is responsive to signals generated by joystick 26 and is programmed to allow a user to move forward or backwards through the displayed text. In particular, processing module 44 retrieves different data segments from RAM 50 in response to movement of the joystick.
Several systems for encoding the data cards are appropriate and have been described in the prior art. For example, in U.S. Pat. No. 6,176,427 there is described a method for coding digital data, such as a text file, into a pattern printable on an A4 or Letter size piece of paper. In the system that is described it is possible to encode slightly more than 1 MB of data on to one side of a printed letter size page of paper using a high resolution printer and a 600 dpi scanner. In the presently described preferred embodiment the scanner head 30 is implemented by means of the scan head technology described in the previously incorporated U.S. patent application Ser. No. 09/113,053 Such a scanner has an output resolution of 4800 dpi.
It is further envisaged that the data card be produced using the very high resolution print heads described in the previously referred to U.S. patent application Ser. No. 09/113,053.
Accordingly the amount of data that may be stored on a data card of dimensions 8.5 cm×5 cm (3.5″×2″) is approximately 1 Mb. Encoding of the text on to the data card may be performed as described in U.S. patent application Ser. No. 09/112,781 which is hereby incorporated by reference in its entirety.
Accordingly an entire novel may be stored on a single credit card sized plastic card by means of a pattern formed as an array of 16 million printed ink dots. The manufacturing cost per card is less than 1 cent, or about one fiftieth the cost of manufacturing a floppy disk. While it is envisaged that the card be made of plastic it would also be possible to use other substrates such as paper.
While it is primarily envisaged that the data stored on the data card will correspond to the text of a book or magazine, it is also possible to encode an executable program file. Accordingly updates to the software program stored in program memory 43 may be conveniently distributed in the form of encoded data cards.
The mechanical arrangement of the e-book will now be described further with reference to FIG. 7 where it will be noted that front door 6 and rear door 8 are independently pivoted about hinges 50 and 52. Power cable 29 is deliberately left slack to accommodate movement of the front door 6 during closure of the book. It will be noted that the spine 16 and outer surfaces of the front and rear doors are configured so that upon fully opening the e-book the flexible LCD screen is drawn taught and flat for convenient viewing.
A further cross sectional view of the e-book, with doors 6 and 8 brought to a closed position appears in FIG. 8. It will be noted that in the closed position a mid portion 54 of the flexible LCD screen 24 is able to loop into the spine by virtue of a recess formed in the spine for and front and rear doors for receiving the screen. Consequently creasing and damage of the LCD screen is avoided.
Also visible in FIG. 8 are screen-to-PCB contact areas 58, 60 which respectively connect the underside of the PCB to the outer edges of each of PCBs 26 and 28. As previously explained, conductive traces on the underside of the PCB provide a path for the PCBs to exchange power and data signals.
A further cross-sectional view is provided in FIG. 9 through the long axis of spine 16 showing two AAA batteries located in a battery compartment formed in the spine.
As will be realized by those skilled in the art, embodiments of the invention other than the preferred embodiment described in detail herein are possible. Accordingly the following claims are not to be read as limited by the preferred embodiment.
the microprocessor circuitry is configured to convert the data signals into image data to be viewed on the screen display.
2. An electronic book as claimed in claim 1, wherein the microprocessor circuitry includes at least one printed circuit board mounted in the housing and loaded with processing modules including a central processor unit, a BIOS memory integrated circuit and a RAM.
3. An electronic book as claimed in claim 2, wherein the housing is foldable about the spine member.
4. An electronic book as claimed in claim 3, wherein the screen display is flexible and mounted on respective inner sides of the housing portions.
5. An electronic book as claimed in claim 4, wherein the screen display is partially received within the cavity of the spine member.
6. An electronic book, as claimed in claim 4, wherein the at least one printed circuit board is connected to the flexible display via conductive traces terminating in peripheral contact regions of the display.
7. An electronic book as claimed in claim 1, wherein a battery compartment is positioned in the spine member, the battery compartment being connected to the at least one printed circuit board to supply power thereto.
8. An electronic book, as claimed in claim 1, further comprising a roller mechanism positioned in the housing proximate the card slot and arranged to retract the card into the housing, the roller mechanism being actuated when a card is inserted into the slot.
9. An electronic book, as claimed in claim 8, further comprising a card insertion sensor positioned in the housing proximate the slot to detect the insertion of the card into the slot and to generate a signal to activate the roller mechanism.
EP0620937A1 (en) 1992-01-10 1994-10-26 Eastman Kodak Co Image-based electronic pocket organizer.


US President Donald Trump reportedly told his aides he never wanted to meet someone as "lifeless" as Nigerian President Muhammadu Buhari again.
Trump's remarks came after Buhari's visit to the White House earlier this year, said a BBC report quoting Financial Times.
"Trump likes chemistry. Africa has never been high on his radar but if the big guy likes you he'll find a way to make things work," an unnamed source reportedly told the Financial Times ahead of a meeting between Trump and Kenyan President Uhuru Kenyatta on Monday.
Buhari was the first sub-Saharan African president to visit Trump at the White House where Trump also reportedly managed to stir a few controversies during the meeting, according to BBC.
During their meeting Trump reportedly asked Buhari what the situation was with "that Boca Haram" referring to Boko Haram and also showed little understanding of the conflict between herdsmen and farmers in Nigeria, owing it only to a religious divide and a Christian genocide.
Trump has also angered many Africans over the past few months with his ignorance as he called some countries - including those on the African continent – "shithole" countries, his mispronunciation of Namibia as "Nambia" and the latest tweet weighing in on the matters of land ownership in South Africa.


There is a change to the Road Map Below. Instead of ERC20-TOKEN now we have upgraded to Waves Wallet for more security and stability.. See this post on Bitcointalk..
Start the first phase of SWAP of 5000000000 SWAP Tokens 1.0 (TOKEN) at the rate of USD 1 for 10000 SWAP TOKEN COIN (STC) which is equivalent to 0.00210785 ETH as per the USD/ETH exchange rate on Yobit as of now. Though we will be using ETH in the future for all financial transactions, for the sake of easier understanding and stability in exchange rates we will do all our calculations and initial transactions based on USD.
To retain any benefit TOKEN investors may get by the price rise in TOKEN on Yobit due to these announcements it has been decided to make the exchange through USD and not through TOKENS. Hence this is more of a sale of SWAPT TOKEN COIN than an exchange of TOKEN. The sale will be done as follows. It will be the choice of TOKEN holders to either hold their TOKEN for profiting from whatever price rise TOKEN will experience over the course of this road map due to these positive developments and sell their TOKEN for profit at a later date or if they feel price may drop, to sell now to generate whatever the amount USD required to buy Swap Token Coins (STC).
*Payment Process for this Phase – Create Yobit Code for 2 USD (as an example) like this below.
(https://metamask.io/#how-it-works) This is an ETH Wallet. You can receive your STC here and store it.
Now combine your Yobit Code and your wallet address with a + sign between the two as shown below.
In the body of the message just give the combined code and your Yobit id as follows.
Once the Yobit code is activated and the payment is received you will receive your 20000 STC (for 2 USD as an example) with a confirmation mail.
To see your balance you have to click on MetaMask extension press the Tokens button in MetaMask and press add token button and paste the following contract address in the Token Contract Address Box and add the token STC.
Your balance will show up immediately. This way you can get the coins without any transaction fees on your part even though the financial value might be as low as 2 USD.
Development of SWAP TOKEN COIN using a forked CryptoNote algorithm. Development of Wallet, Blockchain, Masternodes and setting up of Initial infrastructure.
The algorithm will be based on a proprietary Verified MacID Secured Payment (VMSP) infrastructure and the same technology will be used for (POW) mining as well. Mining can be done only on devices (Android based Pocket Miners that can also be used as Mobile Phones/Tablets) with built-in mining software, sold by STC on our website through an exclusive Amazon Webstore and is secured by the unique MacID (machine ID of the device). More Details about the technology will be revealed in a White Paper to be released by the 15th of April 2018. More detailed information will be released once an application is filed for patenting both the product and the mining process based on the VMSP infrastructure technology.
The profits generated from device sales will be shared via airdrops every quarter to all stake holders in the ratio of 40:40:20 meaning 40% profit shared with stake holders in proportion to the STC held by them as per a quarterly snapshot, 40% for future development and research to take the coin and the business to greater heights and 20% for maintenance of infrastructure and overheads.
SWAP of ECR20 TOKEN Issued to MetaMask with new STC coins based on the forked and released CryptoNote algorithm.
Listing of the NEW STC coin on three exchanges.
Prototype testing of Pocket Miner Devices for mining efficiency and device technical compatibility and stability. Manufacturing tie up for production of Pocket Mining Software and pre-order of Devices on STC website.
Device Release for trial sale and for market acceptance testing through exclusive Amazon Webstore.
Commencement of Regular Sale of Pocket Miner Devices on webstore. Setting up of back end customer care arrangement for technical and software related assistance.
First Airdrop of Quarterly Profits.
Additional research and Development to upgrade the Pocket Miner as a stand-alone device that can be connected to various appliances, automobiles etc., so that customers can generate coins to subsidise their electricity charges, fuel bills, Talk Time Payments, Data and Cable Charges etc., The future is to put one Pocket Miner Device in every single kid's pocket so that even the pocket money and educational needs of kids can be subsidised in a secure manner.
Review of performance and release of new Road Map for the next two years.
STC plans to retain 30 percent of initial sale proceeds as cash as a safety net for investors. What this means is at any point in time if investors for whatever reason wish to sell back the coins after the STC coin Wallet SWAP during the second half of June 2018 they can do so and get a minimum of 30% of their original investment as USD. For example, if an investor has bought USD 200 worth of STC during the original SWAP or during the ICO he or she can sell back the coins for a minimum of USD 60 on the STC website any time after 17th of June 2018. See white paper for more details about the business and financial security for investors.
Why dont yobit providing auto swap?
This is a project taken up separately to support TOKEN holders. It is different from TOKEN and Yobit.